Some Aspects of Moyal Deformed Integrable Systems

TL;DR

This paper explores the mathematical structure of Moyal $ heta$-deformed integrable systems, focusing on their Lax representations for $sl_2$ KdV$_{ heta}$ and Burgers equations, to understand their integrability properties.

Contribution

It introduces a systematic study of integrability in the $ heta$-deformation framework using Moyal momentum algebra and extends Lax representations to deformed integrable systems.

Findings

Lax representations are successfully extended to $ heta$-deformed systems.

Key properties of $ heta$-deformed integrable equations are identified.

The framework provides new insights into the mathematical structure of noncommutative integrable systems.

Abstract

Besides its various applications in string and D-brane physics, the $\theta$-deformation of space (-time) coordinates (naively called the noncommutativity of coordinates), based on the $\star$-product, behaves as a more general framework providing more mathematical and physical informations about the associated system. Similarly to the Gelfand-Dickey framework of pseudo differential operators, the Moyal $\theta$-deformation applied to physical problems makes the study more systematic. Using these facts as well as the backgrounds of Moyal momentum algebra introduced in previous works [21, 25, 26], we look for the important task of studying integrability in the $\theta$-deformation framework. The main focus is on the $\theta$-deformation version of the Lax representation of two principal examples: the $sl_2$ KdV$_{\theta}$ equation and the Moyal $\theta$-version of the Burgers systems. Important properties are presented.